Professor Greenleaf received his AB and SM degrees from the University of Chicago in 1977 and the PhD from Princeton University in 1981. He then spent two years at MIT as an NSF Postdoctoral Fellow, following which he came to the University of Rochester in 1983 as an assistant professor. He was promoted to associate professor in 1986, unlimited tenure in 1989 and full professor in 1997. He spent Fall 1987 at the Mathematical Sciences Research Institute in Berkeley and the 1990-91 academic year at the University of Washington, supported by a Sloan Research Fellowship.
Professor Greenleaf's research interests are in harmonic analysis and microlocal analysis, with applications to integral geometry and inverse problems. In recent years, he has been particularly interested in estimates for oscillatory integral and Fourier integral operators with degenerate phase functions. These arise in looking at solutions to certain partial differential equations, and from averaging operators associated with families of curves or lines in n-space. The latter include X-ray transforms which provide the mathematical underpinnings of CAT scanning. Recently, Professor Greenleaf has also been interested in multiplicative properties of Fourier integral distributions. Controlling these allows one to obtain uniqueness and reconstruction in various inverse problems, such as determining a potential function from the backscattering data ( a subset of the scattering kernel of the associated wave equation) or from the Cauchy data of the associated time-independent Schrödinger equation.
More recently, Professor Greenleaf, together with Matti Lassas of the Helsinki University of Technology, Yaroslav Kurylev of University College, London, and Gunther Uhlmann of the University of Washington, have been using insight gained from the study of inverse problems to give a rigorous foundation and introduce new constructions in the burgeoning field of "cloaking", or invisibility from observation by electromagnetic waves.
Professor Greenleaf's research is supported in part by National Science Foundation grants.